On the metric theory of inhomogeneous Diophantine approximation: An Erd\H{o}s-Vaaler type result

Abstract

In 1958, Sz\"{u}sz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Sz\"{u}sz's theorem states that for any non-increasing approximation function $\psi:\mathbb{N}\to (0,1/2)$ with $\sum_q \psi(q)=\infty$ and any number $\gamma,$ the following set \[ W(\psi,\gamma)=\{x\in [0,1]: |qx-p-\gamma|< \psi(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $\gamma$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition $\psi(q)=O((q(\log\log q)^2)^{-1}).$ In particular, this covers the case when $\gamma$ is not Liouville, for example $\pi,e,\ln 2, \sqrt{2}.$ In general, if $\gamma$ is irrational, $\psi(q)=O(q^{-1}(\log\log q)^{-2})$ and in addition, \[ \left(\liminf_{Q\to\infty} \sum_{q=Q}^{Q^{(\log Q)^{1/8} }}\psi(q)\right)=\infty, \] then $W(\psi,\gamma)$ has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.